Problem 16 - A Graphical Approach to College Algebra

Question

Determine whether each statement is true or false. If is false, tell why. No real number is a pure imaginary number.

Step-by-Step Solution

Answer

The statement is true; no real number can be a pure imaginary number.

Step 1: Define Real and Pure Imaginary Numbers

A real number is any number that can be found on the number line, including all the fractions and irrational numbers such as \(rac{3}{4}\ ext{ or }\sqrt{2}\). A pure imaginary number is written in the form \(bi\ ext{ where }b\text{ is a real number and }i\text{ is the imaginary unit with the property that }i^2 = -1\).

Step 2: Analyze the Statement

The statement claims that no real number is a pure imaginary number. A pure imaginary number is defined by having an imaginary component \((bi)\) and no real component, i.e., there is no real number that can be written in the form \(bi\ ext{ with }b eq 0\).

Step 3: Conclusion on the Truthfulness of the Statement

Since a real number has no imaginary component while a pure imaginary number has no real component, they are distinctly different. This means that no real number can be a pure imaginary number. Therefore, the statement is true.

Key Concepts

Pure Imaginary Numbers

Pure imaginary numbers are fascinating components of the number system. These numbers are expressed in the form of \( bi \), where \( b \) is a real number and \( i \) is the imaginary unit. One of the key aspects of a pure imaginary number is that its real part is equal to zero. This makes them unique from real numbers.

Consider this; if you have \( 0 + 5i \), this number is a pure imaginary number because there is no real number involved in its expression—just the part \( 5i \).

• A pure imaginary number does not appear on the traditional number line since it requires a different dimension for visualization than what real numbers need.
• Understanding pure imaginary numbers is crucial because they allow us to solve equations that we wouldn't be able to with just real numbers.

Imaginary Unit

The imaginary unit, often represented by \( i \), is a cornerstone of working with complex numbers, including pure imaginary numbers. The unique property of the imaginary unit is its definition: \( i^2 = -1 \). This definition is what allows complex numbers to exist.

The introduction of \( i \) gives us a way to deal with the square roots of negative numbers, which is impossible to solve within the realm of real numbers.

• If you try to solve \( x^2 + 1 = 0 \), you can't find a real number solution because the square of a real number is always positive or zero.
• Applying the imaginary unit provides the solution \( x = i \) and \( x = -i \), offering real utility in mathematics and engineering.

Number Line

The number line is an essential visualization tool used to represent real numbers. It stretches in both positive and negative directions infinitely. On this line, every point represents a real number, including integers, fractions, and irrational numbers like \( \sqrt{2} \).

The number line, however, does not include or represent imaginary or pure imaginary numbers because these numbers exist in a different conceptual space entirely.

• When learning about complex numbers, mathematicians visualize them using the complex plane, where the horizontal axis represents real numbers, and the vertical axis represents imaginary numbers.
• This allows us to see numbers like \( 3 + 4i \) plotted in a two-dimensional space, unlike pure imaginary numbers which do not appear on the traditional number line.